This section is intended to introduce various aspects of the art, which may be associated with examples of the disclosed techniques. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the disclosed techniques. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
There is considerable amount of interest in the oil industry and various scientific communities in understanding the underlying physics of sedimentary bodies created by meandering channel systems in the context of oil reservoir formation and geologic modeling. The last decade has witnessed a significant progress in the research and development of evolutionary physics-based sedimentary models, commonly known as “process-based” geological models. One example of such a model is described in U.S. Patent Publication No. 2007/0219725, entitled “A Method For Evaluating Sedimentary Basin Properties By Numerical Modeling Of Sedimentation Processes”, by Sun, et al., filed on Aug. 23, 2005. Process-based models are now capable of generating geologically realistic models by time-advancing the governing equations of fluid flow and sedimentation laws from an initial topography. The process- or physics-based approach limits the number of ad-hoc parameters and complies with fundamental laws of nature in its time evolution.
Although process-based modeling is considered to be a great improvement over purely statistical techniques, one also needs to choose the initial and boundary conditions, initial topography, and other model parameters in a manner that the model prediction corresponds closely to available field data. This is not an easy undertaking and integration of field and production data into hydrodynamics-based models is a formidable task, known as “conditioning” in geology. Conditioning of physics-based sedimentary processes is an inverse problem constrained by partial differential equations of fluid motion and sedimentation. Generally speaking, an inverse problem is a problem in which model parameters are derived from known data. Inverse problems are generally difficult to solve for a variety of reasons and are considerably harder for process-based approaches in sedimentary systems because of the scarcity of field data needed to constrain the process.
To facilitate automated methods for conditioning, inverse problems are usually posed as optimization problems and solved with various known methods devised in the field of optimization. These methods fall in two major groups, namely, gradient-based techniques and direct search methods. Direct search methods rely on forward simulations alone to explore the parameter space for the global extremum while gradient-based methods also need sensitivity information with which to locate a local extremum in the vicinity of an initial estimate. The term parameter space, as used herein, refers to a multi-dimensional vector space that has the same dimension as the number of model parameters. The term forward simulation, as used herein, refers to a numerical solution of the governing physical laws in a given spatial and temporal domain for a set of input parameters representing a point in the parameter space.
Optimal control deals with the problem of finding the unknowns of a model constrained by some known data. In geologic problems such as predicting meander bends, the set of unknowns or model parameters may potentially include but are not limited to spatial and temporal distribution of model parameters, initial terrain topography, boundary conditions on the terrain perimeter, time history of terrain deformation and movement, such as subsidence, or the like.
The optimality criterion is embodied in the definition of an objective function (i.e., cost function) which serves as a quantitative measure of the deviation of the model prediction from observed field data. The specific form of the objective function can have a large influence on the outcome of the approach, and physical understanding is required to help guide the selection of candidate objective functions. Gradient-based approaches require gradient (sensitivity) information to explore the parameter space and find a local extremum. An adjoint model may allow computation of sensitivity information and has the distinctive advantage that the computational cost does not scale with the number of unknown parameters. As such, it can be used with relative ease for large-scale conditioning problems.
A simulation model has been designed for the dynamics of meandering channels which provides a framework for studying the interplay between a migrating channel and the changing sedimentary environment created by the channel itself.
In the course of their evolution, meandering channels can encounter a discontinuous and abrupt process known as a cut-off event. This discontinuity in the forward model greatly affects both the accuracy and stability of the adjoint model. In some cases, the adjoint solution stays stable but the accuracy of the sensitivity information is undermined with some components even having the wrong sign. However, in most cases, the adjoint solution, which runs in reverse time, becomes unstable a few time steps prior to the occurrence of the discontinuous event and no sensitivity information can be computed. A proper treatment of the discontinuity is required to ensure numerical stability of the adjoint model and safeguard the accuracy of the obtained sensitivities.